Residue

 Singularity:


A singularity is a point at which a function becomes undefined, infinite, or behaves in a way that prevents the function from being smooth or continuous.

Isolated Singularity:


An isolated singularity is a singularity that is distinct and separated from other singularities. In other words, it is a point where the function misbehaves, but if you look in a small neighborhood around that point, the function is well-defined and analytic (smooth).

Types of Isolated Singularities:


Removable Singularity: The function can be defined at the singular point in such a way that it becomes analytic in the entire neighborhood. The singularity is "removed" by defining the function appropriately.

Pole: The function approaches infinity as you get closer to the singular point.

Essential Singularity: The function has a more complicated behavior at the singular point, and it cannot be "smoothed out" or made analytic in the neighborhood.

Examples:


For the function 

�

(

�

)

=

1

�

f(z)= 

z

1


 , the point 

�

=

0

z=0 is an isolated singularity, and it is a simple pole.

For the function 

�

(

�

)

=

�

1

�

g(z)=e 

z

1


 

 , the point 

�

=

0

z=0 is an isolated singularity, and it is an essential singularity.

Behavior in a Neighborhood:


The term "isolated" emphasizes that the singularity is not part of a cluster of singularities; it stands alone. The function behaves well in a small region surrounding the isolated singularity.

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